1: Precalculus
- Page ID
- 216145
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Functions
- This page covers the concept of functions as relationships between inputs and outputs, distinguishing between functions and non-functions with examples. It explains function notation, evaluation, solving, and different representations like tables and graphs. The vertical line test is introduced for graphical identification of functions.
- 1.2: Operations on Functions
- This page covers the concepts of composition and transformation of functions. It illustrates how one function's output can serve as another's input, exemplified with cost and temperature functions. Additionally, it details various transformations, including shifts, reflections, and stretches, using specific functions as examples. The importance of the order of transformations is highlighted, particularly in altering function behavior and graphing.
- 1.3: Linear Functions
- This page covers linear functions and their applications in real-life scenarios. It explains the relationship between variables, illustrated by a taxi fare equation, and emphasizes the importance of slope and y-intercept in grappling with linear equations. The content also includes methods for determining rates of change and finding intersections.
- 1.4: Exponents
- This page covers the Laws of Exponents, essential for simplifying algebraic expressions. It outlines key rules for multiplying and dividing powers with the same base, raising powers to powers, and managing products and quotients. It also addresses special cases like zero and negative exponents and introduces fractional exponents as radicals.
- 1.5: Quadratics
- This page covers quadratic functions and their transformations, emphasizing real-world applications in geometry and economics. It explains various forms of quadratic functions, how to find intercepts using the quadratic formula, and includes a projectile motion example.
- 1.6: Polynomials and Rational Functions
- This page provides an overview of polynomial and rational functions, defining key elements like degree, leading term, and leading coefficient. It explains finding intercepts and solving polynomial inequalities, and introduces rational functions with a focus on vertical and horizontal asymptotes. Practical examples illustrate these concepts, including the transformation of reciprocal functions and real-world applications such as distance, speed, and sugar concentration.
- 1.7: Exponential Functions
- This page contrasts linear and exponential growth using examples from two companies, highlighting linear growth with a constant increase versus exponential growth based on percentage changes. It explains the exponential growth function, its applications in population dynamics, finance, and decay processes, and discusses graphing characteristics, including the influence of parameters \( a \) and \( b \) on exponential graphs.
- 1.8: Logarithmic Functions
- This page covers logarithms as inverse functions of exponentials, detailing their properties, including conversions between forms, and the significance of common and natural logs. It emphasizes the shape of logarithmic graphs influenced by their base and the critical domain restriction to positive values.
- 1.E: Review (Exercises)
- This page features a collection of exercises aimed at enhancing understanding of mathematical functions, including their properties, evaluations, and applications in real-life scenarios. Activities cover function notation, composites, linear equations, inequalities, and mathematical modeling, focusing on scenarios like population growth and resource comparisons.